On proper extensions of the conformal group

• Autor(a) principal: Mello, Ulisses Lakatos de
• Data de Publicação: 2022
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45132/tde-13092022-115126/
• topic: Ação de grupos em superfícies; Entropia topológica; Groups acting on surfaces; Grupos topológicos; Ransitividade; Topological entropy; Topological groups; Transitivity

It is proven in this essay that any group of orientation preserving diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. This means that any two ordered lists of four distinct points can be mapped one onto the other via a transformation in the group, isotopic to the identity. In addition, it is shown that any such group must always contain an element of positive topological entropy, for which a description as isotopic to a relative pseudo-Anosov homeomorphism of the 4-punctured sphere is provided. Furthermore, an elementary characterisation of the Möbius transformations within the full group of sphere diffeomorphisms is given in terms of transitivity.